Question

Radioactive waste is packed in a long, thin-walled cylindrical container. The waste generates thermal energy non-uniformly, according to the relationship Qgen=Qo[1-(r-ro)^2], where Qo is a constant and ro is the radius of the container. Steady-state conditions are maintained by submerging the container in a liquid is at T(infinity) and provides a uniform convection coefficient h. Obtain an expression for the temperature Ts of the container wall under steady conditions.

Answer 1

`T_s=T_\infty +(Q_or_o)/(4h)`

Step-by-step explanation:

Heat generated per unit volume

`Q_(gen)=Qo\left [{1-(r^2)/(r_o^2)}\right]`

Total heat generation through the volume

Eg= Q(gen) .dV

`E_g=\int_(0)^(r_o)Qo\left [{1-(r^2)/(r_o^2)}\right]\ 2\pi rLdr`

`E_g=2\pi LQ_o\int_(0)^(r_o)\left [{r-(r^3)/(r_o^2)}\right]dr`

`E_g=2\pi LQ_o\left [(r^2)/(2)-(r^4)/(4r_o^2)\right]^(r_0)_0`

`E_g=(1)/(2)Q_0\pi L\ r_0^2` -------1

Heat transfer due to convection

q= h A (Ts- T ∞) ----------2

By equating equation 1 and 2

`h 2\pi r_0 L(T_s-T_\infty ) =(1)/(2)Q_0\pi L\ r_0^2`

`T_s=T_\infty +(Q_or_o)/(4h)`